Fictionalism

Abstract

This chapter considers how Steiner’s characterisation of the problem of mathematical applicability relates to that of another distinguished philosopher of mathematics, Hartry Field. A key mathematical notion unifying these two very different philosophers in their approach to mathematical applicability is that of homomorphism. Rather than digress to explain the notion in detail at this point, though, explanation for the non-mathematical reader is relegated to an Appendix. The thrust of the narrative and the overview desired requires a detailed look at Field’s work and its relation to Wittgenstein’s. This chapter exposes, amongst important differences between Wittgenstein and Field, some important and illuminating similarities. Each philosopher in his different way turns away from realism in the philosophy of mathematics … and each sees mathematics as in some sense involved with rules of inference rather than assertions. This needs careful unpacking, and this chapter is possibly the most technically demanding part of the book philosophically and logically. A key notion in contemporary philosophy of mathematics is that of indispensability. In short, the idea is that science—particularly contemporary physics—is essentially mathematical; mathematics seems indispensable for science. This has apparent consequences for our philosophical view of mathematics; as Hilary Putnam (one half of the originator of the so-called Quine-Putnam indispensability thesis) put it, ‘… quantification over mathematical entities is indispensable for science…this commits us to accepting the existence of the mathematical entities’. Once we accept that mathematical entities exist, however, we are likely to start worrying about how we can ever get to know anything about such entities, particularly given their abstract, non-causal nature. Of course this is another way of seeing how Einstein’s enigma arises: we are faced with an apparent dilemma (well-known to philosophers of mathematics, often described as Benacerraf’s dilemma) arising from the difficulty—impossibility, Benacerraf claimed—of squaring a demand for uniformity of semantics (demanding acceptance of existence) with an appropriate epistemology (explaining our knowledge) of mathematics. By the end of Chap. 2 the reader will have a clearer view of these matters as well as the beginnings of a serious and solid critique. Hartry Field attempts to blunt the horns of Benacerraf’s dilemma by denying the existence of mathematical entities and the truth of mathematical statements; Wittgenstein can be read as questioning the nature of the uniformity of semantics. Field is a (nominalist) fictionalist about mathematics: mathematics, he says, is a useful fiction. Details of how such fictionalism intersects with Wittgenstein’s philosophy emerge in the chapter under the following schematic plan: fictionalism, applicability, and ‘face value’ … nominalism and indispensability … conservativeness … representation theorems … Field and Wittgenstein: mathematics as rules … semantic uniformity and ‘taking mathematics at face value’ … Benacerraf’s dilemma … two approaches to challenging semantic homogeneity … arguments from Frege, Wright and Field … knowing non-existence … deductivism and implausibility … meanings and beetles: Wittgenstein on semantic homogeneity … the mathematical ‘realm’.